External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid Olivier Glass * ,J´ozsefJ.Kolumb´an † , Franck Sueur ‡ Abstract We consider the motion of a rigid body due to the pressure of a surrounded two-dimensional irrotational perfect incompressible fluid, the whole system being confined in a bounded domain with an impermeable condition on a part of the external boundary. Thanks to an impulsive control strategy we prove that there exists an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the im- mersed rigid body is driven from some given initial position and velocity to some final position and velocity in a given positive time, without touching the exter- nal boundary. The controlled part of the external boundary is assumed to have a nonvoid interior and the final position is assumed to be in the same connected component of the set of possible positions as the initial position. Keywords— Fluid-solid interaction; impulsive control; geodesics; coupled ODE- PDE system; fluid mechanics; Euler equation; control problem; external boundary control. MSC: 76B75, 93C15, 93C20. * CEREMADE, UMR CNRS 7534, Universit´ e Paris-Dauphine, PSL Research University, Place du Mar´ echal de Lattre de Tassigny, 75775 Paris Cedex 16, France † CEREMADE, UMR CNRS 7534, Universit´ e Paris-Dauphine, PSL Research University, Place du Mar´ echal de Lattre de Tassigny, 75775 Paris Cedex 16, France ‡ Institut de Math´ ematiques de Bordeaux, UMR CNRS 5251, Universit´ e de Bordeaux, 351 cours de la Lib´ eration, F33405 Talence Cedex, France. 1
Transcript
External boundary control of the motion of a rigid body
immersed in a perfect two-dimensional fluid
Olivier Glass∗, Jozsef J. Kolumban †, Franck Sueur‡
Abstract
We consider the motion of a rigid body due to the pressure of a surrounded
two-dimensional irrotational perfect incompressible fluid, the whole system being
confined in a bounded domain with an impermeable condition on a part of the
external boundary. Thanks to an impulsive control strategy we prove that there
exists an appropriate boundary condition on the remaining part of the external
boundary (allowing some fluid going in and out the domain) such that the im-
mersed rigid body is driven from some given initial position and velocity to some
final position and velocity in a given positive time, without touching the exter-
nal boundary. The controlled part of the external boundary is assumed to have
a nonvoid interior and the final position is assumed to be in the same connected
component of the set of possible positions as the initial position.
PDE system; fluid mechanics; Euler equation; control problem; external boundary
control.
MSC: 76B75, 93C15, 93C20.
∗CEREMADE, UMR CNRS 7534, Universite Paris-Dauphine, PSL Research University, Place du
Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France†CEREMADE, UMR CNRS 7534, Universite Paris-Dauphine, PSL Research University, Place du
Marechal de Lattre de Tassigny, 75775 Paris Cedex 16, France‡Institut de Mathematiques de Bordeaux, UMR CNRS 5251, Universite de Bordeaux, 351 cours de
la Liberation, F33405 Talence Cedex, France.
1
Contents
1 Introduction and main result 3
1.1 The model without control . . . . . . . . . . . . . . . . . . . . . . . . . 3
The main result of this paper is the following statement.
Theorem 1. Let T > 0. Consider S0 ⊂ Ω bounded, closed, simply connected with
smooth boundary, which is not a disk, and u0 ∈ C∞(F(0);R2), γ ∈ R, q0 = (h0, 0), q1 =
(h1, ϑ1) ∈ Q, h′0, h′1 ∈ R2, ϑ′0, ϑ
′1 ∈ R, such that (h0, 0) and (h1, ϑ1) belong to the same
connected component of Q and
div u0 = curlu0 = 0 in F(0), u0 · n = 0 on ∂Ω,
u0 · n = (h′0 + ϑ′0(x− h0)⊥) · n on ∂S0,
∫∂S0
u0 · τ dσ = γ.
Then there exists a control g ∈ C∞0 ((0, T )×Σ) and a solution (h, ϑ, u) ∈ C∞([0, T ];Q)×C∞([0, T ];C∞(F(t);R2)) to (1.1), (1.2), (1.6), (1.7), (1.8), (1.9), (1.10), (1.11), which
satisfies (h, h′, ϑ, ϑ′)(T ) = (h1, h′1, ϑ1, ϑ
′1).
5
Note that there is a slight abuse of notation in writing C∞([0, T ];C∞(F(t);R2)),
since the domain in which the fluid evolves is also time-dependent.
(h0, 0) (h′1, ϑ′1)
Σ ⊂ ∂Ω
(h′0, ϑ′0)
(h1, ϑ1)
Figure 2: The initial and final positions and velocities in the control problem
Remark 1. In Theorem 1 the control g can be chosen with an arbitrary small total
flux through Σ−, that is for any T > 0, for any ν > 0, there exists a control g and a
solution (h, ϑ, u) satisfying the properties of Theorem 1 and such that moreover∣∣∣∣∫ T
0
∫Σ−
u · n dσdt∣∣∣∣ < ν.
See Section 5.4 for more explanations. Let us mention that such a small flux condition
cannot be guaranteed in the results [7, 14, 16] regarding the controllability of the Euler
equations.
When S0 is a disk, the second equation in (1.6) becomes degenerate, so it needs to
be treated separately. For instance, in the case of a homogeneous disk, i.e. when the
center of mass coincides with the center of the disk and we have (x−h(t))⊥ ·n = 0, for
any x ∈ ∂S(t), t ≥ 0, hence we cannot control ϑ. However, we have a similar result for
controlling the center of mass h.
Theorem 2. Let T > 0. Given a homogeneous disk S0 ⊂ Ω, u0 ∈ C∞(F(0);R2),
γ ∈ R, h0, h1 ∈ Ω, h′0, h′1 ∈ R2, such that (h0, 0) and (h1, 0) are in the same connected
component of Q, and div u0 = curlu0 = 0 in F(0), u0 · n = 0 on ∂Ω, u0 · n = h′0 · non ∂S0,
∫∂S0 u0 · τ dσ = γ, there exists g ∈ C∞0 ((0, T ) × Σ) and a solution (h, u) in
C∞([0, T ]; Ω) × C∞([0, T ];C∞(F(t);R2)) of (1.1), (1.2), (1.6), (1.7), (1.9), (1.10),
(1.11), (1.12) with initial data (h0, h′0, u0), which satisfies (h, h′)(T ) = (h1, h
′1).
The proof is similar to that of Theorem 1, with the added consideration that (x−h(t))⊥ · n = 0, for any x ∈ ∂S(t), t ≥ 0. We therefore omit the proof. In the case
where the disk is non-homogeneous the analysis is technically more intricate already in
the uncontrolled setting, see [21], and we will let aside this case in this paper.
6
References. Let us mention a few results of boundary controllability of a fluid alone,
that is without any moving body. The problem is then finding a boundary control which
steers the fluid velocity from u0 to some prescribed state u1. For the incompressible
Euler equations small-time global exact boundary controllability has been obtained in
[7, 16] in the 2D, respectively 3D case. This result has been recently extended to the case
of the incompressible Navier-Stokes equation with Navier slip-with-friction boundary
conditions in [10], see also [11] for a gentle exposition. Note that the proof there relies
on the previous results for the Euler equations by means of a rapid and strong control
which drives the system in a high Reynolds regime. This strategy was initiated in [8],
where an interior controllability result was already established. For “viscous fluid +
rigid body” control systems (with Dirichlet boundary conditions), local controllability
results have already been obtained in both 2D and 3D, see e.g. [2, 3, 30]. These
results rely on Carleman estimates on the linearized equation, and consequently on the
parabolic character of the fluid equation.
A different type of fluid-solid control result can be found in [22], where the fluid is
governed by the two-dimensional Euler equation. However in this paper the control is
located on the solid’s boundary which makes the situation quite different.
Actually, the results of Theorem 1 and Theorem 2 can rather be seen as some
extensions to the case of an immersed body of the results [17, 18, 19] concerning La-
grangian controllability of the incompressible Euler and Stokes equations, where the
control takes the same form as here.
Generalizations and open problems. First, as we mentioned before, using the
techniques of this paper, the result could be straightforwardly generalized for non simply
connected domains. One could also manage in the same way the control of several solids
(the reader may in particular see that the argument using Runge’s theorem in Section 7
is local around the solid). We would also like to underline that the absence of vorticity
is not central here. This may surprise the reader acquainted with the Euler equation,
but actually following the arguments of Coron [7, 8], one knows how to control the full
model when one can control the irrotational one. This is by the way the technique
that we use to take care of the circulation γ (see in particular Section 3). But the
presence of vorticity makes a lot of complications from the point of view of the initial
boundary problem, in particular for what concerns the uniqueness issue, see Yudovich
[38]. To avoid these unnecessary technical complications, we restrain ourselves to the
irrotational problem. But the full problem could undoubtedly be treated in the same
way.
There remain also many open problems. Considering the recent progresses on the
7
controllability in the viscous case, a natural question is whether or not the results in this
paper could be adapted to the case where a rigid body is moving in a fluid driven by the
incompressible Navier-Stokes equation, with Navier slip-with-friction boundary condi-
tions. We hope that the analysis performed here in the case of “inviscid fluid + rigid
body” control systems could be used in order to get small-time global controllability
results of “viscous fluid + rigid body” control systems.
Let us mention the following open problem regarding the motion planning of a rigid
body immersed in an inviscid incompressible irrotational flow.
Open problem 1. Let T > 0, (h0, 0) in Q, ξ in C2([0, T ];Q), with ξ(0) = (h0, 0).
Let us decompose ξ′(0) into ξ′(0) = (h′0, ϑ′0). Consider S0 ⊂ Ω bounded, closed, simply
connected with smooth boundary, which is not a disk, γ ∈ R, and u0 ∈ C∞(F(0);R2)
such that div u0 = curlu0 = 0 in F(0), u0 ·n = 0 on ∂Ω, u0 ·n = (h′0 +ϑ′0(x−h0)⊥) ·non ∂S0 and
∫∂S0 u0 · τ dσ = γ. Do there exist g ∈ C0([0, T ] × Σ) and a solution
(h, ϑ, u) ∈ C2([0, T ];Q) × C∞([0, T ];C1(F(t);R2)) to (1.1), (1.2), (1.6), (1.7), (1.8),
(1.9), (1.10), (1.11), which satisfies ξ = (h, ϑ)?
Even the approximate motion planning in C2, i.e. the same statement as above but
with ‖ξ − (h, ϑ)‖C2([0,T ]) ≤ ε (with ε > 0 arbitrary) instead of ξ = (h, ϑ), is an open
problem.
Plan of the paper. The paper is organized as follows. In Section 2 we first recall
from [21] a reformulation of the Newton equations (1.6) as an ODE in the uncontrolled
case and then extend it to the case with control. In particular in the case with zero
circulation and no control this ODE is the geodesic equation associated with a metric
which encodes the added mass phenomenon.
In Section 3 we prove that Theorem 1 can be deduced from a simpler result, namely
Theorem 4, where the solid displacement, the initial and final solid velocities and the
circulation are assumed to be small.
In Section 4 we prove that another reduction is possible, as we prove that an ap-
proximate controllability result (rather than an exact one), namely Theorem 5, allows
to deduce Theorem 4.
Section 5 is devoted to the proof of Theorem 5 and is the core of the paper.
In Section 6 we prove a Proposition that is important for Theorem 5, namely that
we can approximate the whole system by a simpler one in a certain regime.
Section 7 explains how one can construct the control by means of complex analysis:
it can be considered as the cornerstone of our control strategy.
8
2 Reformulation of the solid’s equation into an ODE
In this section we establish a reformulation of the Newton equations (1.6) as an ODE
for the three degrees of freedom of the rigid body with coefficients obtained by solving
some elliptic-type problems on a domain depending on the solid position. Indeed the
fluid velocity can be recovered from the solid position and velocity by an elliptic-type
problem, so that the fluid state may be seen as solving an auxiliary steady problem,
where time only appears as a parameter, instead of the evolution equation (1.1). The
Newton equations can therefore be rephrased as a second-order differential equation on
the solid position whose coefficients are determined by the auxiliary fluid problem.
Such a reformulation in the case without boundary control was already achieved in
[21] and we will start by recalling this case in Section 2.1, cf. Proposition 1 below. A
crucial fact in the analysis is that in the ODE reformulation the pre-factor of the body’s
accelerations is the sum of the inertia of the solid and of the so-called “added inertia”
which is a symmetric positive-semidefinite matrix depending only on the body’s shape
and position, and which encodes the amount of incompressible fluid that the rigid body
has also to accelerate around itself. Remarkably enough in the case without control
and where the circulation is 0 it turns out that the solid equations can be recast as a
geodesic equation associated with the metric given by the total inertia.
Then we will extend this analysis to the case where there is a control on a part of the
external boundary in Section 2.2, cf. Theorem 3. In particular we will establish that
the remote influence of the external boundary control translates into two additional
force terms in the second-order ODE for the solid position; indeed we will distinguish
one force term associated with the control velocity and another one associated with its
time derivative.
To simplify notations, we denote the positions and velocities q = (h, ϑ), q′ = (h′, ϑ′),
and
S(q) = h+R(ϑ)(S0 − h0) and F(q) = Ω \ S(q),
since the dependence in time of the domain occupied by the solid comes only from the
position q. Furthermore, we denote q(t) = (h(t), ϑ(t)).
2.1 A reminder of the uncontrolled case
We first recall that in the case without any control the fluid velocity satisfies (1.2),
(1.3), (1.7) and (1.9). Therefore at each time t the fluid velocity u satisfies the following
9
div/curl system:div u = curlu = 0 in F(q),
u · n = 0 on ∂Ω and u · n =(h′ + ϑ′(x− h)⊥
)· n on ∂S(q),∫
∂S(q)u · τ dσ = γ,
(2.1)
where the dependence in time is only due to the one of q and q′. Given the solid position
q and the right hand sides, the system (2.1) uniquely determines the fluid velocity u in
the space of C∞ vector fields on the closure of F(q). Moreover thanks to the linearity
of the system with respect to its right hand sides, its unique solution u can be uniquely
decomposed with respect to the following functions which depend only on the solid
position q = (h, ϑ) in Q and encode the contributions of elementary right hand sides.
• The Kirchhoff potentials
Φ = (Φ1,Φ2,Φ3)(q, ·) (2.2)
are defined as the solution of the Neumann problems
∆Φi(q, x) = 0 in F(q), ∂nΦi(q, x) = 0 on ∂Ω, for i ∈ 1, 2, 3,
∂nΦi(q, x) =
ni on ∂S(q), for i ∈ 1, 2,(x− h)⊥ · n on ∂S(q), for i = 3,
(2.3)
where all differential operators are with respect to the variable x.
• The stream function ψ for the circulation term is defined in the following way.
First we consider the solution ψ(q, ·) of the Dirichlet problem ∆ψ(q, x) = 0 in
F(q), ψ(q, x) = 0 on ∂Ω, ψ(q, x) = 1 on ∂S(q). Then we set
ψ(q, ·) = −
(∫∂S(q)
∂nψ(q, x) dσ
)−1
ψ(q, ·), (2.4)
such that we have ∫∂S(q)
∂nψ(q, x) dσ = −1,
noting that the strong maximum principle gives us ∂nψ(q, x) < 0 on ∂S(q).
Remark 2. The Kirchhoff potentials Φ and the stream function ψ are C∞ as functions
of q on Q. We will use several times some properties of regularity with respect to the
domain of solutions to linear elliptic problems, included for another potential A[q, g]
associated with the control, see Definition 1 below. We will mention along the proof
the properties which will be used and we refer to [6, 28, 32] for more on this material
which is now standard in fluid-structure interaction.
10
The following statement is an immediate consequence of the definitions above.
Lemma 1. For any q = (h, ϑ) in Q, for any p = (`, ω) in R2 × R and for any γ, the
unique solution u in C∞(F(q)) to the following system:div u = curlu = 0 in ∈ F(q),
u · n = 0 on ∂Ω and u · n =(`+ ω(x− h)⊥
)· n on ∂S(q),∫
∂S(q)u · τ dσ = γ.
(2.5)
is given by the following formula, for x in F(q),
u(x) = ∇(p · Φ(q, x)) + γ∇⊥ψ(q, x). (2.6)
Above p · Φ(q, x) denotes the inner product p · Φ(q, x) =∑3
i=1 piΦi(q, x).
Let us now address the solid dynamics. The solid motion is driven by the Newton
equations (1.6) where the influence of the fluid on the solid appears through the fluid
pressure. The pressure can in turn be related to the fluid velocity thanks to the Euler
equations (1.1). The contributions to the solid dynamics of the two terms in the right
hand side of the fluid velocity decomposition formula (2.6) are very different. On the
one hand the potential part, i.e. the first term in the right hand side of (2.6), contributes
as an added inertia matrix, together with a connection term which ensures a geodesic
structure (see [34]), whereas on the other hand the contribution of the term due to the
circulation, i.e. the second term in the right hand side of (2.6), turns out to be a force
which reminds us of the Lorentz force in electromagnetism by its structure (see [21]).
We therefore introduce the following notations.
• We respectively define the genuine and added mass 3× 3 matrices by
Mg =
m 0 0
0 m 0
0 0 J
,
and, for q ∈ Q,
Ma(q) =
(∫F(q)∇Φi(q, x) · ∇Φj(q, x) dx
)16i,j63
.
Note that Ma is a symmetric Gram matrix and is C∞ on Q.
• We define the symmetric bilinear map Γ(q) given by
〈Γ(q), p, p〉 =
∑1≤i,j≤3
Γki,j(q) pi pj
1≤k≤3
∈ R3, ∀p ∈ R3,
11
where, for each i, j, k ∈ 1, 2, 3, Γki,j denotes the Christoffel symbols of the first
kind defined on Q by
Γki,j =1
2
(∂(Ma)k,j
∂qi+∂(Ma)k,i∂qj
− ∂(Ma)i,j∂qk
). (2.7)
It can be checked that Γ is of class C∞ on Q.
• We introduce the following C∞ vector fields on Q with values in R3 by
E = −1
2
∫∂S(q)
|∂nψ(q, ·)|2∂nΦ(q, ·) dσ, (2.8)
B =
∫∂S(q)
∂nψ(q, ·) (∂nΦ(q, ·)× ∂τΦ(q, ·)) dσ. (2.9)
We recall that the notation Φ was given in (2.2).
The reformulation of the model as an ODE is given in the following result, which
was first established in [34] in the case γ = 0 and in [21] in the case γ ∈ R.
Theorem 3. Given q = (h, ϑ) ∈ C∞([0, T ];Q), u ∈ C∞([0, T ];C∞(F(q(t));R2)) we
have that (q, u) is a solution to (1.1), (1.2), (1.3), (1.6), (1.7) and (1.9) if and only if
q satisfies the following ODE on [0, T ](Mg +Ma(q)
Substituting (2.13) into (2.12) and then the resulting decomposition of ∇π into (2.11)
we get
Mg q′′ = −
∫F(q)
(∂t∇(q′ · Φ(q, x)) +
∇|∇(q′ · Φ(q, x))|2
2
)· ∇Φ(q, x) dx
−γ∫F(q)
(∂t∇⊥ψ(q, x) +∇
(∇(q′ · Φ(q, x)) · ∇⊥ψ(q, x)
))· ∇Φ(q, x) dx
−γ2
∫F(q)
∇|∇ψ(q, x)|2
2· ∇Φ(q, x) dx.
(2.14)
According to Lemmas 32, 33 and 34 in [21], the terms in the three lines of the right-
hand side above are respectively equal to −Ma(q)q′′ − 〈Γ(q), q′, q′〉, γq′ × B(q) and
γ2E(q), so that we easily deduce the ODE (2.10) from (2.14).
The conservation of the kinetic energy 12
(Mg+Ma(q)
)q′ ·q′ is then simply obtained
by multiplying the ODE (2.10) by q′ and observing that((Mg +Ma(q)
)q′′ + 〈Γ(q), q′, q′〉
)· q′ =
(1
2
(Mg +Ma(q)
)q′ · q′
)′. (2.15)
2.2 Extension to the controlled case
We now tackle the case where a control is imposed on the part Σ of the external
boundary ∂Ω. At any time this control has to be compatible with the incompressibility
of the fluid meaning that the flux through Σ has to be zero. We therefore introduce
the set
C :=
g ∈ C∞0 (Σ;R) such that
∫Σg dσ = 0
.
The decomposition of the fluid velocity u then involves a new potential term in-
volving the following function.
13
Definition 1. With any q ∈ Q and g ∈ C we associate the unique solution α :=
A[q, g] ∈ C∞(F(q);R) to the following Neumann problem:
∆α = 0 in F(q) and ∂n α = g1Σ on ∂F(q), (2.16)
with zero mean on F(q).
Let us mention that the zero mean condition above allows to determine a unique
solution to the Neumann problem but plays no role in the sequel.
Now Lemma 1 can be modified as follows.
Lemma 2. For any q = (h, ϑ) in Q, for any p = (`, ω) in R2 × R, for any g in C, the
unique solution u in C∞(F(q)) to
div u = curlu = 0 in F(q),
u · n = 1Σ g on ∂Ω and u · n =(`+ ω(x− h)⊥
)· n on ∂S(q),∫
∂S(q)u · τ dσ = γ,
is given by
u = ∇(p · Φ(q, ·)) + γ∇⊥ψ(q, ·) +∇A[q, g]. (2.17)
Let us avoid a possible confusion by mentioning that the ∇ operator above has to
be considered with respect to the space variable x. The function A[q, g] and its time
derivative will respectively be involved into the arguments of the following force terms.
Definition 2. We define, for any q in Q, p in R3, α in C∞(F(q);R) and γ in R,
F1(q, p, γ)[α] and F2(q)[α] in R3 by
F1(q, p, γ)[α] := −1
2
∫∂S(q)
|∇α|2 ∂nΦ(q, ·) dσ (2.18)
−∫∂S(q)
∇α ·(∇(p · Φ(q, ·)) + γ∇⊥ψ(q, ·)
)∂nΦ(q, ·) dσ,
F2(q)[α] := −∫∂S(q)
α∂nΦ(q, ·) dσ. (2.19)
Observe that Formulas (2.18) and (2.19) only require α and ∇α to be defined on
∂S(q). Moreover when these formulas are applied to α = A[q, g] for some g in C, then
only the trace of α and the tangential derivative ∂τα on ∂S(q) are involved, since the
normal derivative of α vanishes on ∂S(q) by definition, cf. (2.16).
We define our notion of controlled solution of the “fluid+solid” system as follows.
Definition 3. We say that (q, g) in C∞([0, T ];Q)×C∞0 ([0, T ]; C) is a controlled solution
if the following ODE holds true on [0, T ]:(Mg +Ma(q)
)q′′ + 〈Γ(q), q′, q′〉 = γ2E(q) + γq′ ×B(q)
+ F1(q, q′, γ)[α] + F2(q)[∂tα],(2.20)
where α(t, ·) := A[q(t), g(t, ·)].
14
We have the following result for reformulating the model as an ODE.
Proposition 1. Given
q ∈ C∞([0, T ];Q), u ∈ C∞([0, T ];C∞(F(q(t));R2)) and g ∈ C∞0 ([0, T ]; C),
we have that (q, u) is a solution to (1.1), (1.2), (1.6), (1.7), (1.8), (1.9), (1.10), (1.11),
(1.12) if and only if (q, g) is a controlled solution and u is the unique solution to the
unique div/curl type problem:
div u = curlu = 0 in F(q),
u · n = 1Σ g on ∂Ω and u · n =(h′ + ϑ′(x− h)⊥
)· n on ∂S(q),∫
∂S(q)u · τ dσ = γ,
with q = (h, ϑ).
Proposition 1 therefore extends Theorem 3 to the case with an external bound-
ary control (in particular one recovers Theorem 3 in the case where g is identically
vanishing).
Proof. We proceed as in the proof of Theorem 3 recalled above, with some modifications
due to the extra term involved in the decomposition of the fluid velocity, compare (2.6)
and (2.17). In particular some extra terms appear in the right hand side of (2.14) after
substituting the right hand side of (2.17) for u in (2.12). Using some integration by
parts and the properties of the Kirchhoff functions we obtain integrals on ∂S(q) whose
sum precisely gives F1(q, q′, γ)[α(t, ·)] + F2(q)[∂tα(t, ·)]. This allows to conclude.
3 Reduction to the case where the displacement, the ve-
locities and the circulation are small
For δ > 0, we introduce the set
Qδ = q ∈ Ω× R : d(S(q), ∂Ω) > δ. (3.1)
The goal of this section is to prove that Theorem 1 can be deduced from the following
result. The balls have to be understood for the Euclidean norm (rather than for the
metric Mg +Ma(q)).
Theorem 4. Given δ > 0, S0 ⊂ Ω bounded, closed, simply connected with smooth
boundary, which is not a disk, q0 in Qδ and T > 0, there exists r > 0 such that for
any q1 in B(q0, r), for any γ ∈ R with |γ| ≤ r and for any q′0, q′1 ∈ B(0, r), there is a
controlled solution (q, g) in C∞([0, T ];Qδ)×C∞0 ([0, T ]×Σ) such that (q, q′)(0) = (q0, q′0)
and (q, q′)(T ) = (q1, q′1).
15
Remark in particular that for r > 0 small enough, B(q0, r) is included in the con-
nected component of Qδ containing q0.
Proof of Theorem 1 from Theorem 4. We proceed in two steps: first we use a time-
rescaling argument in order to deduce from Theorem 4 a more general result covering
the case where the initial and final velocities q′0 and q′1 and the circulation γ are large.
This argument is reminiscent of a time-rescaling argument used by J.-M. Coron for
the Euler equation [7], which has been also used in [22] in order to pass from the
potential case to the case with vorticity. Then we use a compactness argument in order
to deal with the case where q0 and q1 are remote (but of course in the same connected
component of Qδ).The time-rescaling argument relies on the following observation: it follows from
(2.20) that (q, g) is a controlled solution on [0, T ] with circulation γ if and only if
(qλ, gλ) is a controlled solution on [0, λT ] with circulation γλ , where (qλ, gλ) is defined
by
qλ(t) := q
(t
λ
)and gλ(t, x) :=
1
λg
(t
λ, x
). (3.2)
Of course the initial and final conditions
(q, q′)(0) = (q0, q′0) and (q, q′)(T ) = (q1, q
′1)
translate respectively into
(qλ, (qλ)′)(0) =
(q0,
q′0λ
)and (qλ, (qλ)′)(T ) =
(q1,
q′1λ
). (3.3)
Now consider q0 in Qδ and q1 in B(q0, r) in the same connected component of Qδas q0, with r > 0 as in Theorem 4, and q′0, q′1 and γ without size constraint. For λ
small enough, the corresponding (qλ, (qλ)′)(0) and (qλ, (qλ)′)(T ) satisfy the assumptions
of Theorem 4. Hence there exists a controlled solution (q, g) on [0, λT ], achieving
(q, q′)(0) = (q0, q′0) and (q, q′)(λT ) = (q1, q
′1), for λ small enough. Moreover we can
assume that it is the case without loss of generality that λ is small, and in particular
that λ ≤ 1. Thus the result is obtained but in a shorter time interval.
To get to the desired time interval, using that Equation (2.20) enjoys some invari-
ance properties by translation and time-reversal (up to the change of the sign of γ) it
is sufficient to glue together an odd number, say 2N + 1 with N in N∗, of appropriate
controlled solutions each defined on a time interval of length λT with λ = 12N+1 , going
back and forth between (q0, q′0) and (q1, q
′1) until time T = (2N + 1)λT . Moreover one
can see that the gluings are not only C2 but even C∞.
We have therefore already proven that Theorem 1 is true in the case where q1 is
close to q0, or more precisely for any q0 in Qδ and q1 in B(q0, rq0).
16
For the general case where q0 and q1 are in the same connected component of Qδ for
some δ > 0, without the closeness condition, we use again a gluing process. Consider
indeed a smooth curve from q0 to q1. For each point q on this curve, there is a rq > 0
such that for any q in B(q, rq), any q′0, q′1 and any γ, one can connect (q, q′0) to (q, q′1) by
a solution of the system, for any time T > 0. Extract a finite subcover of the curve by
the balls B(q, rq). Therefore we find N ≥ 2 and (q iN
)i=1,...,N−1 in the same connected
component of Qδ as q0 such that for any i = 1, . . . , N , q iN
is in B(q i−1N, rq i−1
N
) (note
that this includes q0 and q1). Therefore, using again the local result obtained above,
there exist some controlled solutions from (q i−1N, 0) to (q i
N, 0) (for i = 1 and i = N
we use (q0,q′0N ) and (q1,
q′1N ) rather than (q0, 0) and (q1, 0)), each on a time interval of
length T associated with circulation γN . One deduces by time-rescaling some controlled
solutions associated with circulation γ on a time interval of length TN . Gluing them
together leads to the desired controlled solution.
4 Reduction to an approximate controllability result
The goal of this section is to prove that Theorem 4 can be deduced from the following
approximate controllability result thanks to a topological argument already used in
[22], see Lemma 3 below. Let us mention that a similar argument has also been used
for control purposes but in other contexts, see e.g. [1, 5, 26, 27].
Theorem 5. Given δ > 0, S0 ⊂ Ω bounded, closed, simply connected with smooth
boundary, which is not a disk, q0 in Qδ and T > 0, there is r > 0 such that B(q0, r)
is included in the same connected component of Qδ as q0 and such that for any γ ∈ Rwith |γ| ≤ r and for any q′0 in B(0, r), for any η > 0, there is a mapping
T : B((q0, q
′0), r
)→ C∞([0, T ];Qδ)
which to (q1, q′1) associates q where (q, g) is a controlled solution associated with the
initial data (q0, q′0), such that the mapping
(q1, q′1) ∈ B
((q0, q
′0), r
)7→(T (q1, q
′1), T (q1, q
′1)′)(T ) ∈ Qδ × R3
is continuous and such that for any (q1, q′1) in B
((q0, q
′0), r
),
‖(T (q1, q
′1), T (q1, q
′1)′)(T )− (q1, q
′1)‖ 6 η.
The proof of Theorem 5 will be given in Section 5. Here we prove that Theorem 4
follows from Theorem 5.
Proof of Theorem 4 from Theorem 5. Let δ > 0, S0 ⊂ Ω bounded, closed, simply con-
nected with smooth boundary, which is not a disk, q0 in Qδ and T > 0. Let r > 0
17
as in Theorem 5. Let γ ∈ R with |γ| ≤ r and q′0 in B(0, r). According to Theorem 5
applied with η = r2 there is a mapping T : B
((q0, q
′0), r
)→ C∞([0, T ];Qδ) which maps
(q1, q′1) to q where (q, g) is a controlled solution associated with the initial data (q0, q
′0),
such that for any (q1, q′1) in B
((q0, q
′0), r
), ‖(T (q1, q
′1), T (q1, q
′1)′)(T ) − (q1, q
′1)‖ 6 r
2 .
We define a mapping f from B((q0, q
′0), r
)to R6 which maps (q1, q
′1) to f(q1, q
′1) :=(
T (q1, q′1), T (q1, q
′1)′)(T ). Then we apply the following lemma borrowed from [22, pages
32-33], to w0 = (q0, q′0) and κ = r.
Lemma 3. Let w0 ∈ Rn, κ > 0, f : B(w0, κ) → Rn a continuous map such that we
have |f(w)− w| ≤ κ2 for any x in ∂B(w0, κ). Then B(w0,
κ2 ) ⊂ f(B(w0, κ)).
This allows to conclude the proof of Theorem 4 setting r = κ2 = r
2 .
5 Proof of the approximate controllability result Theo-
rem 5
In this section we prove Theorem 5 by exploiting the geodesic feature of the uncontrolled
system with zero circulation, cf. the observation below Theorem 3. To do so, we will
use some well-chosen impulsive controls which allow to modify the velocity q′ in a short
time interval and put the state of the system on a prescribed geodesic (and use that
|γ| is small). We mention here [4] and the references therein for many more examples
on the impulsive control strategy.
5.1 First step
We consider S0 ⊂ Ω as before and consider δ > 0 so that q0 ∈ Qδ. We let r1 > 0 be
small enough so that B(q0, r1) ⊂ Qδ. We also let T > 0.
The first step consists in considering the geodesics associated to the uncontrolled,
potential case (γ = 0). The following classical result regarding the existence of geodesics
can be found for instance in [33, Section 7.5], see also [12] for the continuity feature.
Lemma 4. There exists r2 in (0, 12r1) such that for any q1 in B(q0, r2) there exists a
unique C∞ solution q(t) lying in B(q0,12r1) to(
Mg +Ma(q))q′′ + 〈Γ(q), q′, q′〉 = 0 on [0, T ], with q(0) = q0, q(T ) = q1. (5.1)
Furthermore the map q1 ∈ B(q0, r2) 7→ (c0, c1) ∈ R6 given by c0 = q′(0), c1 = q′(T ) is
continuous.
Let us fix r2 as in the lemma before. Let q′0 in B(0, r2) and (q1, q′1) in B
((q0, q
′0), r2
).
18
Our goal is to make the system follow approximately such a geodesic q which we
consider fixed during this Section. For the geodesic equation in (5.1), q0 and q1 deter-
mine the initial and final velocities (which of course differ in general from q′0 and q′1).
But we will see that is possible to use the penultimate term of (2.20) in order to modify
the initial and final velocities of the system. Precisely, the control will be used so that
the right hand side of (2.20) behaves like two Dirac masses at time close to 0 and T ,
driving the velocity q′ from the initial and final velocities to the ones of the geodesic in
two short time intervals close to 0 and T .
5.2 Illustration of the method on a toy model
Let us illustrate this strategy on a toy model. We will later on adapt the analysis to
the complete model, cf. Proposition 4.
Let β : R → R be a smooth, non-negative function supported in [−1, 1], such
that∫ 1−1 β(t)2 dt = 1 and, for ε in (0, 1), βε(t) := 1√
εβ(t−εε
), so that2 (β2
ε )ε is an
approximation of the unity when ε→ 0+.
For a function f defined on [0, T ], we will denote
‖f‖T,ε := ‖f‖C0([0,T ]) + ‖f‖C1([2ε,T−2ε]). (5.2)
Lemma 5. Let q0, r2, q1, q′0 and q′1 as above. Let
v0 :=(Mg +Ma(q0)
)(c0(q1)− q′0) and v1 := −
(Mg +Ma(q1)
)(c1(q1)− q′1). (5.3)
Let, for ε in (0, 1), qε the maximal solution to the following Cauchy problem:(Mg +Ma(qε)
)q′′ε + 〈Γ(qε), q
′ε, q′ε〉 = β2
ε (·) v0 + β2ε (T − ·)v1, (5.4)
with qε(0) = q0 and q′ε(0) = q′0. Then for ε small enough, qε(t) lies in B(q0, r1) for t in
[0, T ] and, as ε→ 0+, ‖qε − q‖T,ε → 0 and (qε, q′ε)(T )→ (q1, q
′1).
Proof. For ε in (0, 1), let us denote Tε = sup T > 0 such that qε(t) ∈ B(q0, r1) for t ∈(0, T ). Let us first prove that there exists T > 0 such that for any ε in (0, 1), Tε ≥ T .
Using the identity (2.15), we obtain indeed, for any ε in (0, 1), for any t ∈ (0, Tε),(Mg+Ma(qε(t))
)q′ε(t)·q′ε(t) =
(Mg+Ma(q0)
)q′0 ·q′0 +2
∫ t
0
(β2ε (·) v0 +β2
ε (T−·)v1
)·q′ε,
Moreover, relying on Remark 2, we see that there exists c > 0 (which depends on δ)
such that for any q in Qδ, for any p in R3,
c|p|2 ≤(Mg +Ma(q)
)p · p ≤ c−1|p|2. (5.5)
2In the next lemma we are going to make use only of the square function β2ε but we will also have
to deal with the function βε itself in the sequel, see below Proposition 2.
19
Therefore using Gronwall’s lemma we obtain that there exists C > 0 such that for any
ε in (0, 1), for any t ∈ (0, Tε), supt∈(0,Tε) ‖q′ε(t)‖ ≤ C. Therefore by the mean value
theorem for T := r1/2C, one has for any ε in (0, 1), Tε ≥ T .
We now prove in the same time that for ε > 0 small enough, Tε ≥ T , and the
convergence results stated in Lemma 5. In order to exploit the supports of the functions
βε(·) and βε(T−·) in the right hand side of the equation (5.4) we compare the dynamics
of qε and q during the three time intervals [0, 2ε], [2ε, T − 2ε] and [T − 2ε, T ].
For ε1 := T /2 and ε in (0, ε1), one already has that Tε ≥ 2ε and we can therefore
simply compare the dynamics of qε and q on the first interval [0, 2ε]. Indeed using again
the mean value theorem we obtain that supt∈[0,2ε] |qε − q0| converges to 0 as ε goes to
0. Moreover integrating the equation (5.4) on [0, 2ε] and taking into account the choice
of v0 in (5.3), we obtain(Mg +Ma(qε(2ε))
)q′ε(2ε) =
(Mg +Ma(q0)
)c0(q1)
−∫ 2ε
0
(DMa(qε) · q′ε
)· q′ε dt−
∫ 2ε
0〈Γ(qε), q
′ε, q′ε〉 dt, (5.6)
Now, there exists C > 0 such that for any q in Qδ, for any p in R3,
|(DMa(q) · p
)· p|+ |〈Γ(q), p, p〉| ≤ C|p|2. (5.7)
Combining this and the bound on q′ε we see that the two terms of the last line of (5.6)
above converge to 0 as ε goes to 0. Since q 7→ Ma(q) is continuous on Qδ and qε(2ε)
converges to q0 as ε→ 0, the matrixMa(qε) converges toMa(q0) as ε→ 0. Therefore,
using that the matrix Mg +Ma(q0) is invertible we deduce that q′ε(2ε) converges to
c0(q1) as ε goes to 0.
During the time interval [2ε, T − 2ε], the right hand side of the equation (5.4)
vanishes and the equation therefore reduces to the geodesic equation in (5.1). Since
this equation is invariant by translation in time, one may use the following elementary
result on the continuous dependence on the data, with a time shift of 2ε.
Lemma 6. There exists η > 0 such that for any (q0, q′0) in B((q0, c0(q1)), η) there exists
a unique C∞ solution q(t) lying in B(q0, r1) to(Mg +Ma(q)
)q′′ + 〈Γ(q), q′, q′〉 = 0
on [0, T ], with q(0) = q0, q′(0) = q′0. Furthermore ‖q − q‖C1([0,T ]) → 0 as (q0, q
′0) →
(q0, c0(q1)).
Since qε(2ε) and q′ε(2ε) respectively converge to q0 and c0(q1), according to Lemma 6
there exists ε2 in (0, ε1) such that for ε in (0, ε2), there exists a unique C∞ solution
qε(t) lying in B(q0, r1) to(Mg +Ma(qε)
)q′′ε + 〈Γ(qε), q
′ε, q′ε〉 = 0 on [0, T ], with qε(0) =
qε(2ε), q′ε(0) = q′ε(2ε) and ‖qε − q‖C1([0,T ]) → 0 as ε→ 0.
Since the function defined by qε(t) = qε(t + 2ε) also satisfies(Mg +Ma(qε)
)q′′ε +
〈Γ(qε), q′ε, q′ε〉 = 0 on [0, T − 4ε], with qε(0) = qε(2ε), q
′ε(0) = q′ε(2ε), by the uniqueness
20
part in the Cauchy-Lipschitz theorem one has that Tε ≥ T − 2ε and qε and qε coincide
on [0, T − 4ε], so that, shifting back in time, ‖qε − q(· − 2ε)‖C1([2ε,T−2ε]) → 0 as ε→ 0.
Since q is smooth, this entails that ‖qε − q‖C1([2ε,T−2ε]) → 0 as ε→ 0.
Finally one deals with the time interval [T −2ε, T ] in the same way as the first step.
In particular, reducing ε one more time if necessary one obtains, by an energy estimate,
a Gronwall estimate and the mean value theorem, that Tε ≥ T . Moreover the choice
of the vector v1 in (5.3) allows to reorientate the velocity q′ε from c1(q1) to q′1 whereas
the position is not much changed (due to the uniform bound of q′ε and the mean value
theorem) so that the value of qε at time T converges to q1 as ε goes to 0.
5.3 Back to the complete model
Now in order to mimic the right hand side of (5.4) we are going to use one part of the
force term F1 introduced in Definition 2. Let us therefore introduce some notations for
the different contributions of the force term F1. We define, for any q in Q, p in R3, α
For what concerns F2 we have, using the property (5.13),
F2(qε,0)[∂tA[qε,0, gε]
]= βε
∫∂S(qε,0)
(DqA[qε,0, g(q0,−2v0)] · q′ε,0
)∂nΦ(qε,0, ·) dσ
+β′ε
(∫∂S(qε,0)
A[qε,0, g(q0,−2v0)] ∂nΦ(qε,0, ·) dσ−∫∂S(q0)
A[q0, g(q0,−2v0)] ∂nΦ(q0, ·) dσ).
Using that the mapping q 7→∫∂S(q)∇qA[q, g(q0, v)] ⊗ ∂nΦ(q, ·) dσ is bounded for
q over Qδ and that the mapping q 7→∫∂S(q)A[q, g(q0, v)] ∂nΦ(q, ·) dσ is Lipschitz with
26
respect to q in Qδ, both uniformly for v in bounded sets of R3, we see that this involves
(recalling the expression of βε given at the beginning of Section 5.2)
∣∣F2(qε,0)[∂tA[qε,0, gε]
]∣∣ . C
(1
ε1/2|q′ε,0|+
1
ε3/2|qε,0 − q0|
), (6.5)
uniformly for (q1, q′1) in B
((q0, q
′0), r2
). Then, multiplying (6.4) by q′ε,0 and using once
more the identity (2.15), we obtain, for any ε in (0, 1), for t in [0,min(2ε, Tε,0)],(Mg +Ma(qε,0(t))
)q′ε,0(t) · q′ε,0(t) =
(Mg +Ma(q0)
)q′0 · q′0
+2
∫ t
0
(F1,a(qε,0)
[A[qε,0, gε]
]+F1,b(qε,0, q
′ε,0)[A[qε,0, gε]
]+F2(qε,0)
[∂tA[qε,0, gε]
])·q′ε,0,
(6.6)
Then, using (5.5), the boundedness of the mappings q 7→ F1,a(q)[A[q, g(q0, v)]] and
q 7→ F1,b(q, ·)[A[q, g(q0, v)]] already mentioned above, the definition of βε and the bound
(6.5), we get
|q′ε,0(t)|2 ≤ C(
1 +1
ε1/2
∫ t
0|q′ε,0(s)|2 ds+
1
ε3/2
∫ t
0|q′ε,0(s)||qε,0(s)− q0| ds
).
Then using the mean value theorem and that t ≤ 2ε, we have that
|q′ε,0(t)|2 ≤ C
(1 + ε1/2 sup
[0,min(2ε,Tε,0)]|q′ε,0|2
),
so that for ε small enough, and for t in [0,min(2ε, Tε,0)], |q′ε,0(t)| ≤ C, uniformly for
(q1, q′1) in B
((q0, q
′0), r2
). As a consequence of the usual blow-up criterion for ODEs,
we have that Tε,0 ≥ 2ε.
During the next phase, i.e. for t in [2ε, T − 2ε], the control is inactive so that the
equation (6.4) is a geodesic equation. Then by a simple energy estimate we get again
that |q′ε,0(t)| ≤ C on [0,min(T − 2ε, Tε,0)].
Finally if Tε,0 ≥ T − 2ε, then we deal with the last phase as in the first phase. This
concludes the proof of Lemma 7.
We then conclude the proof of Proposition 5 by a classical comparison argument
using Gronwall’s lemma and the Lipschitz regularity with respect to q of the various
mappings involved (Ma, Γ, F1,a, F1,b and F2). This allows to prove that there exists
ε2 in (0, ε1] such that for any ε ∈ (0, ε2], Tε,0 ≥ T and ‖qε − qε,0‖C1([0,T ]) → 0 when
ε→ 0+, uniformly for (q1, q′1) in B
((q0, q
′0), r2
). This ends the proof of Proposition 5.
6.4 Proof of Proposition 6
First we may extend Lemma 7 to the solutions qε,γ to (5.16) in the following manner.
27
Lemma 8. There exists εb in (0, ε2) such that ‖q′ε,γ‖C([0,Tε,γ ]) is bounded uniformly in
ε ∈ (0, εb], for any γ ∈ [−1, 1], and for (q1, q′1) ∈ B((q0, q
′0), r2).
It is indeed a matter of adding the “electric field” E in (6.6), and noting that E is
bounded on Qδ; the “magnetic field” B gives no contribution to the energy.
We now finish the proof of Proposition 6. Using a comparison argument we obtain
that there exists ε3 in (0, εb] such that for all ε ∈ (0, ε3], there exists γ0 > 0 such that for
any γ ∈ [−γ0, γ0], we have Tε,γ ≥ T and ‖qε,γ − qε,0‖C1[0,T ] converges to 0 when γ → 0,
uniformly for (q1, q′1) in B
((q0, q
′0), r2
). This concludes the proof of Proposition 6.
7 Design of the control according to the solid position.
Proof of Proposition 2
This section is devoted to the proof of Proposition 2.
7.1 The case of a homogeneous disk
Before proving Proposition 2 we establish the following similar result concerning the
simpler case where the solid is a homogeneous disk. In that case, the statement merely
considers q of the form q = (h, 0). Thus in order to simplify the writing, we introduce
Qhδ := h ∈ R2 such that (h, 0) ∈ Qδ.
Also in all this section when we will write q, it will be understood that q is associated
with h by q = (h, 0).
Proposition 7. Let δ > 0. Then there exists a continuous mapping g : Qhδ × R2 → Csuch that the function α := A[q, g(q, v)] in C∞(F(q);R) satisfies:
∆α(q, x) = 0 in F(q), and ∂nα(q, x) = 0 on ∂F(q) \ Σ, (7.1)∫∂S(q)
|∇α(q, x)|2 ndσ = v, (7.2)∫∂S(q)
α(q, x)ndσ = 0. (7.3)
In order to prove Proposition 7, the mapping g will be constructed using a combi-
nation of some elementary functions which we introduce in several lemmas.
To begin with, we will make use of the elementary geometrical property that
n(q0, x) : x ∈ ∂S(q0) is the unit circle S1 and of the following lemma.
28
Lemma 9. There exist three vectors e1, e2, e3 ∈ n(q0, x) : x ∈ ∂S(q0) and positive
C∞ maps (µi)16i63 : R2 → R+ such that for any v ∈ R2,
3∑i=1
µi(v)ei = v. (7.4)
Proof. One may consider for instance e1 := (1, 0), e2 := (0, 1), e3 := (−1,−1), and
µ1(v) = v1 +√
1 + |v1|2 + |v2|2, µ2(v) = v2 +√
1 + |v1|2 + |v2|2
and µ3(v) =√
1 + |v1|2 + |v2|2.
In the next lemma, we introduce some functions that are defined in a neighbourhood
of ∂S(q0) (for some q0 = (h0, 0) fixed), satisfying some counterparts of the properties
(7.1) and (7.2).
Lemma 10. There exist families of functions (αi,jε )ε∈(0,1), i, j ∈ 1, 2, 3, such that
for any i, j ∈ 1, 2, 3, for any ε ∈ (0, 1), αi,jε is defined and harmonic in a closed
neighbourhood V i,jε of ∂S(q0), satisfies ∂nαi,jε = 0 on ∂S(q0), and moreover one has for
any i, j, k, l in 1, 2, 3,∫∂S(q0)
∇αi,jε · ∇αk,lε ndσ → δ(i,j),(k,l) ei as ε→ 0+.
Proof. Without loss of generality, we may suppose that S(q0) is the unit disk. Consider
the parameterisation c(s) = (cos(s), sin(s)), s ∈ [0, 2π] of ∂S(q0) and the correspond-
ing si such that n(q0, c(si)) = ei, i ∈ 1, 2, 3.We consider families of smooth functions βi,jε : [0, 2π]→ R, i, j ∈ 1, 2, 3, ε ∈ (0, 1),
such that supp βi,jε ∩ supp βk,lε = ∅ whenever (i, j) 6= (k, l), diam(
supp βi,jε)→ 0 as
ε→ 0+,∫ 2π
0βi,jε (s) dσ = 0 and
∣∣∣∣∫ 2π
0|βi,jε (s)|2 n(q0, c(s))ds− ei
∣∣∣∣→ 0 as ε→ 0+.
Then we define αi,jε in polar coordinates as the truncated Laurent series:
αi,jε (r, θ) :=1
2
∑0<k≤K
1
k
(rk +
1
rk
)(−bi,jk,ε cos(kθ) + ai,jk,ε sin(kθ)),
where ai,jk,ε and bi,jk,ε denote the k-th Fourier coefficients of the function βi,jε . It is elemen-
tary to check that the function αi,jε satisfies the required properties for an appropriate
choice of K.
29
Now, for any h ∈ Qhδ , we may define V i,jε (q) := V i,jε −h0+h, which is a neighborhood
of ∂S(q), and αi,jε (q, x) := αi,jε (x + h0 − h), for each x ∈ V i,jε (q). We have for i, j, k, l
in 1, 2, 3,∫∂S(q)
∇αi,jε (q, x) · ∇αk,lε (q, x)n(q, x) dσ =
∫∂S(q0)
∇αi,jε (x) · ∇αk,lε (x)n(q0, x) dσ.
Proceeding as in [14] (see also [13, p. 147-149]) and relying in particular Runge’s theo-
rem, we have the following result which asserts the existence of harmonic approximate
extensions on the whole fluid domain.
Lemma 11. There exists a family of functions (αi,jη )η∈(0,1), i, j ∈ 1, 2, 3, harmonic
in F(q), satisfying ∂nαi,jη (q, x) = 0 on ∂F(q) \ Σ, with for any k in N,
‖αi,jη (q, ·)− αi,jε (q, ·)‖Ck(Vi,jε (q)∩F(q))
→ 0 when η → 0+. (7.5)
We now check that the above construction can be made continuous in q.
Lemma 12. For any ν > 0, there exist continuous mappings h ∈ Qhδ 7→ αi,j(q, ·) ∈C∞(F(q)) where q = (h, 0), i, j ∈ 1, 2, 3, such that for any h ∈ Qhδ , ∆xα
i,j(q, x) = 0
in F(q), ∂nαi,j(q, x) = 0 on ∂F(q) \ Σ and∣∣∣∣∣
∫∂S(q)
∇αi,j(q, ·) · ∇αk,l(q, ·)ndσ − δ(i,j),(k,l) ei
∣∣∣∣∣ ≤ ν. (7.6)
Proof. Let us assume that the functions αi,jη were previously defined not only for h ∈ Qhδbut for h ∈ Qhδ ; this is possible by using a smaller δ. Hence we may for each h ∈ Qhδfind functions αi,jη (for some η > 0) satisfying the properties above, and in particular
such that (7.6) is valid.
Next we observe that for any h ∈ Qhδ , setting q = (h, 0), the unique solution
αi,jη (q, q, ·) (up to an additive constant) to the Neumann problem ∆xαi,jη (q, q, x) = 0 in
F(q), ∂nαi,jη (q, q, x) = 0 on ∂F(q) \ Σ, ∂nα
i,jη (q, q, x) = ∂nα
i,jη (q, x) on Σ, is continuous
with respect to q ∈ Qδ. It follows that when a family of functions αi,jη satisfies (7.6) at
some point h ∈ Qhδ , it satisfies (7.6) (with perhaps 2ν in the right hand side) in some
neighborhood of h. Since Qhδ is compact and can be covered with such neighborhoods,
one can extract a finite subcover and use a partition of unity (according to the variable
q) adapted to this subcover to conclude: one gets an estimate like (7.6) with Cν on
the right hand side (for some constant C). It is then just a matter of considering ν/C
rather than ν at the beginning.
Finally our basic bricks to prove Proposition 7 are given in the following lemma,
where we can add the constraint (7.3).
30
Lemma 13. For any ν > 0, there exist continuous mappings q = (h, 0) ∈ Qδ 7→αi(q, ·) ∈ C∞(F(q)), i ∈ 1, 2, 3, such that for any q = (h, 0) ∈ Qδ, ∆xα
i(q, x) = 0 in
F(q), ∂nαi(q, x) = 0 on ∂F(q) \ Σ and∣∣∣∣∣
∫∂S(q)
∇αi(q, ·) · ∇αj(q, ·)ndσ − δi,j ei
∣∣∣∣∣ ≤ ν, (7.7)∫∂S(q)
αi(q, ·)ndσ = 0. (7.8)
Proof. Consider the functions αi,j given by Lemma 12. For any q = (h, 0) ∈ Qδ, for
any i ∈ 1, 2, 3, the three vectors∫∂S(q) α
i,j(q, ·)ndσ, where j ∈ 1, 2, 3, are linearly
dependent in R2; therefore there exists λi,j(q) ∈ R such that
3∑j=1
λi,j(q)
∫∂S(q)
αi,j(q, ·)ndσ = 0 and
3∑j=1
|λi,j(q)|2 = 1, (7.9)
Then one defines αi(q, ·) :=∑3
j=1 λi,j(q)αi,j(q, ·), and one checks that it satisfies (7.7)
with some Cν in the right hand side. Again changing ν in ν/C allows to conclude.
We are now in position to prove Proposition 7.
Proof of Proposition 7. Let δ > 0. Let ν > 0. We define the mapping S which to
(h, v) ∈ Qhδ × R2 associates the function
α(q, ·) :=3∑i=1
√µi(v)αi(q, ·),
in C∞(F(q)), where the functions µi were introduced in Lemma 9 and the functions αi
were introduced in Lemma 13. Next we define T : Qhδ × R2 → Qhδ × R2 by
(h, v) 7→ (T1, T2)(h, v) :=
(h,
∫∂S(q)
|∇α(q, ·)|2 ndσ
), where α = S(h, v).
Using (7.4) and (7.7), one checks that T is smooth and that
∂T2
∂v= Id +O(ν).
Hence taking ν sufficiently small, we see that ∂T2∂v is invertible, hence ∂T
∂(h,v) is invertible.
Consequently one can use the inverse function theorem on T : for each h0 ∈ Qhδ it realizes
a local diffeomorphism at(h0, 0), and hence on Qhδ × B(0, r) for r > 0 small enough.
This gives the result of Proposition 7 for v small: given (h, v) ∈ Qhδ × B(0, r), we let
(h, v) := T −1(h, v). Then the functions α :=∑3
i=1
√µi(v)αi(q, ·) and g := 1Σ ∂nα
satisfy the requirements. The general case follows by linearity of (7.1) and (7.3) and
by homogeneity of (7.2). This ends the proof of Proposition 7.
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7.2 The case when S0 is not a disk
We now get back to the proof of Proposition 2. We will denote by coni(A) the conical
hull of A, namely
coni(A) :=
k∑i=1
λiai, k ∈ N∗, λi ≥ 0, ai ∈ A
,
The first step is the following elementary geometric lemma.
Lemma 14. Let S0 ⊂ Ω bounded, closed, simply connected with smooth boundary,
which is not a disk. Then coni(n(x), (x− h0)⊥ · n(x)), x ∈ ∂S0 = R3.
Proof. Suppose the contrary. Then there exists a plane separating (in the large sense)
the origin in R3 from the set coni((n(x), (x− h0)⊥ · n(x)), x ∈ ∂S0). We claim that
a normal vector to this plane can be put in the form (a, b, 1), with a, b ∈ R. Indeed,
otherwise it would need to be of the form (a, b, 0), and the separation inequality would
give (a, b) · n(x) ≥ 0, ∀x ∈ ∂S0. However, since ∂S0 is a smooth, closed curve, the set
n(x) : x ∈ ∂S0 is the unit circle of R2, therefore we have a contradiction.
Now we deduce that we have the following separation property:
(a, b) · n(x) + (x− h0)⊥ · n(x) ≥ 0, ∀x ∈ ∂S0.
Denoting w = (a, b) − h⊥0 , this translates into (w + x⊥) · n(x) ≥ 0. But using Green’s
formula, we get
0 ≤∫∂S0
(w + x⊥) · n(x) dσ =
∫S0
div(w + x⊥) dx = 0,
and consequently, we deduce that (w+x⊥) ·n(x) = 0 for all x in ∂S0. This is equivalent
to (x− w⊥) · τ(x) = 0 for all x in ∂S0. Parameterizing the translated curve ∂S0 − w⊥
by c(s), s ∈ [0, 1], it follows that c(s) · c(s) = 0, for all s in [0, 1], and therefore
|c(s)|2 is constant. This means that ∂S0 − w⊥ is a circle, so S0 is a disk, which is a
contradiction.
Fix q0 ∈ Qδ. Recalling the definitions of the Kirchhoff potentials in (2.2) and (2.3),
we infer from the previous lemma that
coni∂nΦ(q0, x), x ∈ ∂S0 = R3.
In place of Lemma 9, we have the following lemma which is a straightforward conse-
quence of Lemma 14 and of a repeated application of Caratheodory’s theorem on the
convex hull.
Lemma 15. There are some (xi)i∈1,...,16 in ∂S0 and positive continuous mappings
µi : R3 → R, 1 6 i 6 16, v 7→ µi(v) such that∑16
i=1 µi(v)∂nΦ(q0, xi) = v.
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We are now in position to establish Proposition 2. We deduce from Lemma 15 that
for any q := (h, ϑ) ∈ Qδ, for any v in R3,
16∑i=1
µi(R(ϑ)v) ∂nΦ(q, xi(q)) = R(ϑ)v,
where xi(q) := R(ϑ)(xi − h0) + h and R(ϑ) denotes the 3 × 3 rotation matrix defined
by
R(ϑ) :=
(R(ϑ) 0
0 1
).
Due to the Riemann mapping theorem, there exists a biholomorphic mapping Ψ :
C\B(0, 1)→ C\S(q) with ∂S(q) = Ψ(∂B(0, 1)), where C denotes the Riemann sphere.
We consider the parametrisations c(s) = (cos(s), sin(s)), s ∈ [0, 2π] of ∂B(0, 1),
respectively Ψ(c(s)), s ∈ [0, 2π] of ∂S(q), and the corresponding si such that xi(q) =
Ψ(c(si)), for i ∈ 1, . . . , 16.Then, for any smooth function α : ∂S(q) → R, due to the Cauchy-Riemann rela-
tions, we have the following:
∂nα(Ψ(x)) =1√
|det(DΨ(x))|∂nB (α Ψ)(x),∫
∂S(q)|∇α(x)|2 ∂nΦ(q, x) dσ =
∫∂B(0,1)
|∇α(Ψ(x))|2 ∂nBΦ(q,Ψ(x))1√
|det(DΨ(x))|dσ,
for any x ∈ ∂B(0, 1), where n and nB respectively denote the normal vectors on ∂S(q)
and ∂B(0, 1). Note that, since Ψ is invertible, we have |det(DΨ(x))| > 0, for any
x ∈ ∂B(0, 1).
For each ε > 0, i ∈ 1, . . . , 16, j ∈ 1, 2, 3, 4 (here the index j belongs to 1, 2, 3, 4rather than 1, 2, 3 in order to adapt the linear dependence argument of Lemma 13
to the case of the three linear constraints (5.13)), we consider families of smooth
